CN104408723B - Raman spectrum image demixing method based on nonnegative matrix approximation - Google Patents
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Abstract
The invention discloses a Raman spectrum image demixing method based on nonnegative matrix approximation. Non-pure pixels do not exist in a Raman spectrum image, unknown limitation on ANC, ASC and end elements exists in Raman spectrum demixing, various regularization factors such as sparsity are added in the nonnegative matrix approximation under the condition that the Raman spectrum image meets requirements of an LMM (labor market model), so that mixed pixel components and corresponding abundance thereof can be obtained effectively. Compared with the existing other method, the Raman spectrum image demixing method based on the nonnegative matrix approximation has the following advantages that the number of end elements (substances) is not required to serve as a priori knowledge; an iteration strategy is adopted, and a gain result is not relevant to the number of the end elements; and a graph relation and 1/2 bound norm are used, so that the noise immunity and the stability of an algorithm are improved.
Description
Technical field
The invention belongs to technical field of image processing, and in particular to a kind of Raman spectral image closed on based on nonnegative matrix
Solution mixing method.
Background technology
Raman spectrum is a kind of 3-D view, including ordinary two dimensional plane picture information and wavelength information.To target
While space characteristics are imaged, tens are formed through dispersion to each space pixel or even hundreds of narrow-band is continuous to carry out
Spectrum cover.One Raman image is the three-dimensional Raman image being made up of the corresponding two dimensional image of several wavelength.
Raman spectrum because of it for different material can all produce the spectral characteristic of uniqueness, and the cell engineering that is widely used
Industries such as (cell wall structure component detections).But current most of Raman images are all by multiple different materials (end member)
Mixing synthesis, in order to more accurately each blending constituent is analyzed, it is necessary to Raman spectral image is carried out to solve mixed analysis,
Generally need to assume that Raman spectral image meets linear mixed model (LMM), the end member abundance in the model needs to meet non-negative
(ANC) and and for 1 restriction (ASC).Under normal circumstances, solution sneaks out journey including two steps of Endmember extraction and Fengdu inverting.
For Endmember extraction, measure of supervision and non-supervisory method are can be largely classified into.Measure of supervision assumes that all of end member is all
It is known, it is main to include fixed point component analyses, automatic Endmember extraction, pure pixel index and iteration error analysis, these methods
Mainly it is analyzed from geometry visual angle, but said method necessarily requires to need at least the presence of an end member in the solid.When
Without in the case of pure end member in algorithm, minimum volume conversion and its similar method (iteration limit end member) are taken comprising institute
There is the maximum simplex of data.Must there is N-1 end member (N is end member sum) in being limited in that of this method, but
It is this to assume undesirable in the data set of real high mixing.After all of Endmember extraction, generally using the full minimum for limiting
Two take advantage of prediction or maximum likelihood analysis to carry out abundance inverting to end member.When end member and its corresponding enriching do not know, EO-1 hyperion
The mixed problem of solution can just regard Blind Signal Separation problem as, and common method includes independent main constituent and nonnegative matrix point
Analysis.It is separate unrealistic in real image between its desired end member for independent principal component method.
The content of the invention
For the above-mentioned technical problem existing for prior art, the invention provides a kind of drawing closed on based on nonnegative matrix
Graceful spectrum picture solution mixing method, adds various regularization factors such as sparse, such that it is able to effectively by closing on to nonnegative matrix
Obtain the component and its corresponding abundance of mixed pixel.
A kind of Raman spectral image solution mixing method closed on based on nonnegative matrix, is comprised the steps：
(1) noise reduction is carried out to Raman spectral image and deblooming is processed；
(2) for any pixel in image after process, the weight between the pixel and other pixels is calculated, so as to obtain
Weight matrix W；
(3) according to described weight matrix W, the regularization factors P of figure relation is calculated；
(4) model is mixed to the solution of Raman spectral image and introduces figure relation regularization factors P and the rarefaction factor, obtain as follows
The mixed Optimized model of solution：
Wherein：F (u, v, Γ) is the Lagrangian with regard to u, v and Г, u for Raman spectral image basic element to
Amount, v is vectorial for the remnants of Raman spectral image, and M is the Raman spectral image after processing, | | | |FFor F- norms, Tr () expressions
The mark of matrix, T representing matrix transposition, Г is Lagrangian matrix, and μ and λ is Lagrangian,It is just with φ
Then change parameter；ГijFor the i-th row jth column element value of Lagrangian matrix, ()ijThe i-th row jth for () interior matrix is arranged
Element value, i and j are natural number and 1≤i≤n, 1≤j≤n, n are the number of pixels of Raman spectral image；
(5) by updating Raman spectral image M and Lagrangian λ, circulate and the mixed Optimized model of above-mentioned solution is minimized
Solve m time, obtain the corresponding basic element vector u of each material in Raman spectral image, and then merging obtains Raman spectral image
Abundance matrix U；M is the species number of material in Raman spectral image.
In described step (2), the weight between pixel and pixel is calculated according to following formula, so as to obtain weight matrix
W；
Wherein：miAnd mjAfter respectively processing in Raman spectral image ith pixel and j-th pixel wavelength-vector, σ
For the core width of heat kernel function, WijWeigh for the weight in Raman spectral image after process between ith pixel and j-th pixel
I-th row jth column element value of weight matrix W.
In described step (3), the regularization factors P of figure relation is calculated according to following formula：
P=Tr (uTLu)
Wherein：L=D-W, D are diagonal matrix and its i-th row the i-th column element valueWijFor weight matrix W's
I-th row jth column element value.
In described step (5), minimum solution is carried out to the mixed Optimized model of solution by following iterative equation group：
Wherein：ukAnd uk-1The basic element of respectively kth time and -1 iteration of kth is vectorial, vkAnd vk-1Respectively kth is secondary
With the remnants vectors of -1 iteration of kth, ГkAnd Гk-1The respectively Lagrangian square of kth time and -1 iteration of kth
Battle array .* and ./respectively represent point multiplication operation and point division operation, ZkAnd Zk-1The respectively middle square of kth time and -1 iteration of kth
Battle array, k is iterationses.
In described step (5), Raman spectral image M and Lagrangian λ are updated according to following formula：
Wherein：MtAnd Mt-1The Raman spectral image for minimizing in solution procedure for respectively the t time and the t-1 time, ut-1With
vt-1Minimize for respectively the t-1 time and solve the remaining vector of the basic element vector sum for obtaining, λtSolved for the t time minimum
Lagrangian in journey,For Raman spectral image MtPixel value composition on l-th wavelength of middle all pixels correspondence
N-dimensional vector, t and l are natural number and 1 ＜ t≤m, 1 ＜ l≤L, and L is the wavelength dimension of Raman spectral image.
The present invention does not exist for the non-pure pixel in Raman spectral image, ANC, the ASC and end during Raman spectrum solution is mixed
The unknown restriction of unit, meets under LMM model cases in Raman spectral image, adds sparse by closing on (NMU) to nonnegative matrix
Etc. various regularization factors, such that it is able to effectively obtain the component and its corresponding abundance of the pixel of mixing.It is of the invention false
If the pixel in Raman spectrum meets linear model, it is proposed that the mixed algorithm of brand-new non-supervisory solution.In in order to meet phenomenon model
Nonnegative value limit, the present invention be based on non-negative algorithm；In order that similar pixel still possesses similar characteristic in decomposition, this
It is bright to propose figure relation for defining the similarity of pixel；In order to remove the noise and noise of Raman spectrum, the present invention is used
Wavelet reconstruction and Savitzky-Golay second order filters；In order to preferably represent the sparse characteristic of data, the present invention is used
L1/2Norm substitutes traditional L0、L1And L2Norm.
Compared with existing additive method, the present invention has the advantage that：1) make without the need for the number of end member (component)
For priori；2) iterative strategy is taken, asks for result unrelated with end member number；3) figure relation and 1/2 norm are added, is improved
The noise immunity and stability of algorithm.
Description of the drawings
Fig. 1 is the schematic flow sheet of Raman spectral image solution mixing method of the present invention.
Specific embodiment
In order to more specifically describe the present invention, below in conjunction with the accompanying drawings and specific embodiment is to technical scheme
It is described in detail.
Embodiment 1：
The present invention is based on order 1-NMU algorithm, it is proposed that the nonnegative matrix of figure relation rarefaction closes on algorithm.
It is shown below, basic order 1-NMU algorithm can be expressed as optimization problem：
Wherein：For F norms.MatrixFor image data matrix, matrix u and matrix v point
Not Wei length be m and length for n row vector.
As shown in figure 1, the Raman spectral image solution mixing method that the present invention is closed on based on nonnegative matrix, comprises the steps：
(1) noise reduction is carried out to Raman image and deblooming is processed, the view data after being processed.
(2) for the pixel in view data, the weight between pixel is calculated according to such as following formula：
Wherein：miAnd mjFor any two pixel in view data, σ is the core width of heat kernel function.
(3) it is shown below, calculates figure relation regularization factors：
Wherein, Tr () is the mark of matrix, and D is diagonal matrix Dii=ΣjWijAnd L=D-W.
(4) based on figure relation and various regularizations, the mixed problem of Raman spectrum solution can be converted to what optimization was shown below
The mixed model of solution：
Wherein：μ, λ are Lagrangian.For above formula, u and v are entered using single step more New Policy
Row estimation, by successive ignition, obtains the decomposition result after an order.Then, using M ← max (0 ,-(M-ukvk T)) using right
M is updated, and continues to be repeated once the decomposition result after order.Until order reaches target sizes, stop iteration.
Minimum solution is carried out to the mixed model of above-mentioned solution, following steps are specifically included：
The determination of 4.1 Lagrangians
In the mixed model of solution, it is thus necessary to determine that tri- Lagrangians of μ, Γ, λ.μ is used to control smoothness, and its value can be with
Determine in an experiment.Γ can be according to Γ(k+1)←max(0,Γ(k)+ak(uvT- M)) it is updated, wherein k is current iteration
Number of times.For λ, can be updated according to following formula.
The more New Policy of 4.2u and v
In the mixed model of solution, the more New Policy for considering u and v in an iterative process, in the present embodiment, Wo Mentong are needed
Cross KKT conditions to solve model, obtain the single step more New Policy being shown below.
V=v.*ZTu./uvT v
Subsequent processing steps after the mixed algorithm of 4.3 solutions
After algorithm stops, view data M is decomposed into U and V by present embodiment.Wherein matrix U be abundance matrix, V by
In error diffusion reason, it is impossible to be taken as end member matrix.In order to obtain abundance matrix and end member matrix.
For abundance matrix, due to must being fulfilled for LMM models in normalization limit, therefore for basis matrix U,
Using such as following formula to its normalization.In order to avoid the appearance of null value, we with the addition of minimum ω during normalization.
Wherein：For the reflected value of basis j in pixel i.
Known image data matrix M and abundance matrix U, we are obtained by the non-negative least square problem solved such as following formula
End member matrix V.
The stopping strategy of 4.4 algorithms
It is the key factor for affecting present embodiment execution performance to stop strategy.In the present embodiment, we are default most
The marginal value that little residual error stops as algorithm, in addition, we arrange the marginal value that maximum iteration time stops as algorithm,
Two kinds of marginal values meet any one, and this algorithm stops.
Embodiment 2：
As a example by below we are using the constituent of tea plant cell wall, method proposed by the present invention is embodied as
Mode is described.
Instrument prepares：
Experimental facilitiess by electronic computer, confocal Raman spectra microscope, Nikon immersion fibroids, it is seen that spectrum ripple
The laser of section, spectral accuracy is～4cm-1, laser voltage is 10mw.
Material prepares：
1, fresh Xihu Longjing Tea blade is gathered in experiment greenhouse, is washed, be placed on laboratory platform；To the Dragon Well tea page
Section is made, spectral region 579.335cm is extracted-1～3062.07cm-1。
Raman image pretreatment：
To Raman spectrum picture, the cell compartment of the area-of-interest (25 × 25) that cutting size is.In order to ensure what is tested
Accuracy, eliminates wave band (1751.2cm-1～3062.07cm-1,1450.7～cm- affected by transparent utensil
11471.4cm-1and 1517.8cm-1～1576.8cm-1), the corresponding clear Raman spectrogram of 462 light waves is chosen altogether
Picture.
Pending Raman image is the high spectrum image that m wavelength size is in the present embodiment, and image size is N=25
×25.The Raman spectrum solution mixing method that the present embodiment is closed on based on nonnegative matrix, it realizes that process is as follows：
1) removal of noise and autofluorescence
In order to remove noise and autofluorescence, ' db1 ' wavelet basiss that we used in Wavelet transformation divide wavelength signals
Solution to 6 layers, only retains 6 layers of high-frequency signal, and original signal is reconstructed using the signal, effectively eliminates substrate and makes an uproar
Impact of the sound to image, then,, using wavelet reconstruction and Savitzky-Golay second order filters, width is 7 pixels for we
Width is filtered to signal, can effectively eliminate interference of the autofluorescence to picture signal.
2) selection of Lagrangian and initiation parameter
In this example, it is thus necessary to determine that tri- Lagrangians of μ, Γ, λ.μ is used to control smoothness, in instances,
μ ∈ (0,0.4) in carry out many experiments with 0.05 as step-length, finally show that, when μ=0.1, algorithm can get optimal value.Γ can
With according to Γ(k+1)←max(0,Γ(k)+ak(uvT- M)) it is updated, wherein k is current iterationses.Renewal for λ
It is in the same manner as in Example 1.In initialization procedure, we using singular value decomposition to data matrix M by being obtained for first
The u and v of secondary iteration.
3) the stopping strategy of algorithm
It is the key factor for affecting this example execution performance to stop strategy.In the present embodiment, we preset Minimum Residual
Difference is 0.001, and in addition, we arrange the marginal value that maximum iteration time stops as algorithm, in this experiment, by reality
Test analysis to learn, when maximum iteration time is 100, algorithm has been restrained.
It is determined to collecting tea cell wall construction composition using present embodiment, can be distinguished by cell wall structure
In each composition distribution and each composition weight.
The above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art
Member, without departing from the inventive concept of the premise, can also make some improvements and modifications, and these improvements and modifications also should be regarded as
In the scope of the present invention.
Claims (5)
1. a kind of Raman spectral image solution mixing method closed on based on nonnegative matrix, is comprised the steps：
(1) noise reduction is carried out to Raman spectral image and deblooming is processed；
(2) for any pixel in image after process, the weight between the pixel and other pixels is calculated, so as to obtain weight
Matrix W；
(3) according to described weight matrix W, the regularization factors P of figure relation is calculated；
(4) model is mixed to the solution of Raman spectral image and introduces figure relation regularization factors P and the rarefaction factor, obtain following solution mixed
Optimized model：
Wherein：F (u, v, Γ) is the Lagrangian with regard to u, v and Г, and u is vectorial for the basic element of Raman spectral image, v
For the remnants vectors of Raman spectral image, M is the Raman spectral image after processing, | | | |FFor F- norms, Tr () representing matrix
Mark, T representing matrix transposition, Г is Lagrangian matrix, and μ and λ is Lagrangian,Regularization ginseng is with φ
Number；ГijFor the i-th row jth column element value of Lagrangian matrix, ()ijFor the i-th row jth column element value of () interior matrix,
I and j are natural number and 1≤i≤n, 1≤j≤n, n are the number of pixels of Raman spectral image；
(5) by updating Raman spectral image M and Lagrangian λ, circulate and solution m is minimized to the mixed Optimized model of above-mentioned solution
It is secondary, the corresponding basic element vector u of each material in Raman spectral image is obtained, and then merge the abundance for obtaining Raman spectral image
Matrix U；M is the species number of material in Raman spectral image.
2. Raman spectral image solution mixing method according to claim 1, it is characterised in that：In described step (2), according to
Following formula calculates the weight between pixel and pixel, so as to obtain weight matrix W；
Wherein：miAnd mjAfter respectively processing in Raman spectral image ith pixel and j-th pixel wavelength-vector, σ is heat
The core width of kernel function, WijIt is weight square for the weight in Raman spectral image after process between ith pixel and j-th pixel
The i-th row jth column element value of battle array W.
3. Raman spectral image solution mixing method according to claim 1, it is characterised in that：In described step (3), according to
Following formula calculates the regularization factors P of figure relation：
P=Tr (uTLu)
Wherein：L=D-W, D are diagonal matrix and its i-th row the i-th column element valueFor weight matrix W's
I-th row jth column element value.
4. Raman spectral image solution mixing method according to claim 1, it is characterised in that：In described step (5), pass through
Following iterative equation group carries out minimum solution to the mixed Optimized model of solution：
Wherein：ukAnd uk-1The basic element of respectively kth time and -1 iteration of kth is vectorial, vkAnd vk-1Respectively kth time and the
The remnants vectors of k-1 iteration, ГkAnd Гk-1The respectively Lagrangian matrix of kth time and -1 iteration of kth .*
With ./point multiplication operation and point division operation, Z are represented respectivelykAnd Zk-1The respectively intermediary matrix of kth time and -1 iteration of kth, k is
Iterationses；D is diagonal matrix and its i-th row the i-th column element valueWijFor the i-th row jth of weight matrix W
Column element value.
5. Raman spectral image solution mixing method according to claim 1, it is characterised in that：In described step (5), according to
Following formula is updated to Raman spectral image M and Lagrangian λ：
Wherein：MtAnd Mt-1The Raman spectral image for minimizing in solution procedure for respectively the t time and the t-1 time, ut-1And vt-1Point
Wei not minimize for the t-1 time and solve the remaining vector of the basic element vector sum for obtaining, λtMinimize in solution procedure for the t time
Lagrangian,For Raman spectral image MtThe n of the pixel value composition on middle all pixels l-th wavelength of correspondence tie up to
Amount, t and l is natural number and 1 ＜ t≤m, 1 ＜ l≤L, and L is the wavelength dimension of Raman spectral image.
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